LMFDB - Embedded newform 1932.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,2,Mod(1,1932)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1932.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1932.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$15.4270976705$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.68740$ of defining polynomial
Character	$\chi$	$=$	1932.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.68740 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.68740 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.22212 q^{13} +2.68740 q^{15} -1.90952 q^{17} +3.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +2.22212 q^{25} +1.00000 q^{27} +6.35375 q^{29} +9.28432 q^{31} +1.00000 q^{33} -2.68740 q^{35} +11.7285 q^{37} -1.22212 q^{39} -10.1938 q^{41} -0.596916 q^{43} +2.68740 q^{45} +1.09048 q^{47} +1.00000 q^{49} -1.90952 q^{51} +0.222116 q^{53} +2.68740 q^{55} +3.00000 q^{57} +14.9717 q^{59} -9.50643 q^{61} -1.00000 q^{63} -3.28432 q^{65} -1.13163 q^{67} +1.00000 q^{69} -8.41595 q^{71} -5.90952 q^{73} +2.22212 q^{75} -1.00000 q^{77} -2.53472 q^{79} +1.00000 q^{81} +10.3537 q^{83} -5.13163 q^{85} +6.35375 q^{87} -0.152683 q^{89} +1.22212 q^{91} +9.28432 q^{93} +8.06220 q^{95} -0.0904840 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - q^{13} + q^{15} + 4 q^{17} + 9 q^{19} - 3 q^{21} + 3 q^{23} + 4 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 3 q^{33} - q^{35} + 6 q^{37} - q^{39} + 3 q^{41} + 15 q^{43} + q^{45} + 13 q^{47} + 3 q^{49} + 4 q^{51} - 2 q^{53} + q^{55} + 9 q^{57} + 14 q^{59} - 2 q^{61} - 3 q^{63} + 14 q^{65} + 9 q^{67} + 3 q^{69} + 11 q^{71} - 8 q^{73} + 4 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 16 q^{83} - 3 q^{85} + 4 q^{87} + 11 q^{89} + q^{91} + 4 q^{93} + 3 q^{95} - 10 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - q^13 + q^15 + 4 * q^17 + 9 * q^19 - 3 * q^21 + 3 * q^23 + 4 * q^25 + 3 * q^27 + 4 * q^29 + 4 * q^31 + 3 * q^33 - q^35 + 6 * q^37 - q^39 + 3 * q^41 + 15 * q^43 + q^45 + 13 * q^47 + 3 * q^49 + 4 * q^51 - 2 * q^53 + q^55 + 9 * q^57 + 14 * q^59 - 2 * q^61 - 3 * q^63 + 14 * q^65 + 9 * q^67 + 3 * q^69 + 11 * q^71 - 8 * q^73 + 4 * q^75 - 3 * q^77 - 12 * q^79 + 3 * q^81 + 16 * q^83 - 3 * q^85 + 4 * q^87 + 11 * q^89 + q^91 + 4 * q^93 + 3 * q^95 - 10 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.68740	1.20184	0.600921	−	0.799309i	$-0.294801\pi$
$5$	2.68740	1.20184	0.600921	+	0.799309i	$0.294801\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−1.22212	−0.338954	−0.169477	−	0.985534i	$-0.554208\pi$
$13$	−1.22212	−0.338954	−0.169477	+	0.985534i	$0.554208\pi$
$14$	0	0
$15$	2.68740	0.693884
$16$	0	0
$17$	−1.90952	−0.463126	−0.231563	−	0.972820i	$-0.574384\pi$
$17$	−1.90952	−0.463126	−0.231563	+	0.972820i	$0.574384\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.22212	0.444423
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.35375	1.17986	0.589931	−	0.807454i	$-0.299155\pi$
$29$	6.35375	1.17986	0.589931	+	0.807454i	$0.299155\pi$
$30$	0	0
$31$	9.28432	1.66751	0.833756	−	0.552133i	$-0.186186\pi$
$31$	9.28432	1.66751	0.833756	+	0.552133i	$0.186186\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−2.68740	−0.454253
$36$	0	0
$37$	11.7285	1.92816	0.964081	−	0.265610i	$-0.0855732\pi$
$37$	11.7285	1.92816	0.964081	+	0.265610i	$0.0855732\pi$
$38$	0	0
$39$	−1.22212	−0.195695
$40$	0	0
$41$	−10.1938	−1.59201	−0.796004	−	0.605291i	$-0.793057\pi$
$41$	−10.1938	−1.59201	−0.796004	+	0.605291i	$0.793057\pi$
$42$	0	0
$43$	−0.596916	−0.0910288	−0.0455144	−	0.998964i	$-0.514493\pi$
$43$	−0.596916	−0.0910288	−0.0455144	+	0.998964i	$0.514493\pi$
$44$	0	0
$45$	2.68740	0.400614
$46$	0	0
$47$	1.09048	0.159063	0.0795317	−	0.996832i	$-0.474658\pi$
$47$	1.09048	0.159063	0.0795317	+	0.996832i	$0.474658\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.90952	−0.267386
$52$	0	0
$53$	0.222116	0.0305100	0.0152550	−	0.999884i	$-0.495144\pi$
$53$	0.222116	0.0305100	0.0152550	+	0.999884i	$0.495144\pi$
$54$	0	0
$55$	2.68740	0.362369
$56$	0	0
$57$	3.00000	0.397360
$58$	0	0
$59$	14.9717	1.94915	0.974576	−	0.224059i	$-0.0719308\pi$
$59$	14.9717	1.94915	0.974576	+	0.224059i	$0.0719308\pi$
$60$	0	0
$61$	−9.50643	−1.21717	−0.608587	−	0.793487i	$-0.708263\pi$
$61$	−9.50643	−1.21717	−0.608587	+	0.793487i	$0.708263\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−3.28432	−0.407369
$66$	0	0
$67$	−1.13163	−0.138251	−0.0691254	−	0.997608i	$-0.522021\pi$
$67$	−1.13163	−0.138251	−0.0691254	+	0.997608i	$0.522021\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−8.41595	−0.998789	−0.499395	−	0.866375i	$-0.666444\pi$
$71$	−8.41595	−0.998789	−0.499395	+	0.866375i	$0.666444\pi$
$72$	0	0
$73$	−5.90952	−0.691657	−0.345828	−	0.938298i	$-0.612402\pi$
$73$	−5.90952	−0.691657	−0.345828	+	0.938298i	$0.612402\pi$
$74$	0	0
$75$	2.22212	0.256588
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−2.53472	−0.285178	−0.142589	−	0.989782i	$-0.545543\pi$
$79$	−2.53472	−0.285178	−0.142589	+	0.989782i	$0.545543\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.3537	1.13647	0.568236	−	0.822866i	$-0.307626\pi$
$83$	10.3537	1.13647	0.568236	+	0.822866i	$0.307626\pi$
$84$	0	0
$85$	−5.13163	−0.556604
$86$	0	0
$87$	6.35375	0.681193
$88$	0	0
$89$	−0.152683	−0.0161843	−0.00809217	−	0.999967i	$-0.502576\pi$
$89$	−0.152683	−0.0161843	−0.00809217	+	0.999967i	$0.502576\pi$
$90$	0	0
$91$	1.22212	0.128113
$92$	0	0
$93$	9.28432	0.962739
$94$	0	0
$95$	8.06220	0.827164
$96$	0	0
$97$	−0.0904840	−0.00918726	−0.00459363	−	0.999989i	$-0.501462\pi$
$97$	−0.0904840	−0.00918726	−0.00459363	+	0.999989i	$0.501462\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−2.66635	−0.265312	−0.132656	−	0.991162i	$-0.542350\pi$
$101$	−2.66635	−0.265312	−0.132656	+	0.991162i	$0.542350\pi$
$102$	0	0
$103$	5.35375	0.527521	0.263760	−	0.964588i	$-0.415037\pi$
$103$	5.35375	0.527521	0.263760	+	0.964588i	$0.415037\pi$
$104$	0	0
$105$	−2.68740	−0.262263
$106$	0	0
$107$	3.22212	0.311494	0.155747	−	0.987797i	$-0.450222\pi$
$107$	3.22212	0.311494	0.155747	+	0.987797i	$0.450222\pi$
$108$	0	0
$109$	−13.4370	−1.28703	−0.643515	−	0.765433i	$-0.722525\pi$
$109$	−13.4370	−1.28703	−0.643515	+	0.765433i	$0.722525\pi$
$110$	0	0
$111$	11.7285	1.11322
$112$	0	0
$113$	11.1316	1.04718	0.523588	−	0.851972i	$-0.324593\pi$
$113$	11.1316	1.04718	0.523588	+	0.851972i	$0.324593\pi$
$114$	0	0
$115$	2.68740	0.250601
$116$	0	0
$117$	−1.22212	−0.112985
$118$	0	0
$119$	1.90952	0.175045
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−10.1938	−0.919147
$124$	0	0
$125$	−7.46528	−0.667715
$126$	0	0
$127$	−1.06220	−0.0942549	−0.0471274	−	0.998889i	$-0.515007\pi$
$127$	−1.06220	−0.0942549	−0.0471274	+	0.998889i	$0.515007\pi$
$128$	0	0
$129$	−0.596916	−0.0525555
$130$	0	0
$131$	0.555767	0.0485576	0.0242788	−	0.999705i	$-0.492271\pi$
$131$	0.555767	0.0485576	0.0242788	+	0.999705i	$0.492271\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	2.68740	0.231295
$136$	0	0
$137$	−9.56863	−0.817503	−0.408752	−	0.912646i	$-0.634036\pi$
$137$	−9.56863	−0.817503	−0.408752	+	0.912646i	$0.634036\pi$
$138$	0	0
$139$	−20.7697	−1.76166	−0.880831	−	0.473430i	$-0.843016\pi$
$139$	−20.7697	−1.76166	−0.880831	+	0.473430i	$0.843016\pi$
$140$	0	0
$141$	1.09048	0.0918353
$142$	0	0
$143$	−1.22212	−0.102199
$144$	0	0
$145$	17.0751	1.41801
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−18.5476	−1.51948	−0.759738	−	0.650229i	$-0.774673\pi$
$149$	−18.5476	−1.51948	−0.759738	+	0.650229i	$0.774673\pi$
$150$	0	0
$151$	8.72855	0.710319	0.355160	−	0.934806i	$-0.384427\pi$
$151$	8.72855	0.710319	0.355160	+	0.934806i	$0.384427\pi$
$152$	0	0
$153$	−1.90952	−0.154375
$154$	0	0
$155$	24.9507	2.00409
$156$	0	0
$157$	1.84008	0.146855	0.0734273	−	0.997301i	$-0.476606\pi$
$157$	1.84008	0.146855	0.0734273	+	0.997301i	$0.476606\pi$
$158$	0	0
$159$	0.222116	0.0176150
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	3.41595	0.267558	0.133779	−	0.991011i	$-0.457289\pi$
$163$	3.41595	0.267558	0.133779	+	0.991011i	$0.457289\pi$
$164$	0	0
$165$	2.68740	0.209214
$166$	0	0
$167$	−10.3748	−0.802826	−0.401413	−	0.915897i	$-0.631481\pi$
$167$	−10.3748	−0.802826	−0.401413	+	0.915897i	$0.631481\pi$
$168$	0	0
$169$	−11.5064	−0.885110
$170$	0	0
$171$	3.00000	0.229416
$172$	0	0
$173$	14.6591	1.11451	0.557256	−	0.830341i	$-0.311854\pi$
$173$	14.6591	1.11451	0.557256	+	0.830341i	$0.311854\pi$
$174$	0	0
$175$	−2.22212	−0.167976
$176$	0	0
$177$	14.9717	1.12534
$178$	0	0
$179$	10.3255	0.771761	0.385881	−	0.922549i	$-0.373898\pi$
$179$	10.3255	0.771761	0.385881	+	0.922549i	$0.373898\pi$
$180$	0	0
$181$	6.65911	0.494968	0.247484	−	0.968892i	$-0.420396\pi$
$181$	6.65911	0.494968	0.247484	+	0.968892i	$0.420396\pi$
$182$	0	0
$183$	−9.50643	−0.702736
$184$	0	0
$185$	31.5193	2.31734
$186$	0	0
$187$	−1.90952	−0.139638
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	3.84008	0.277859	0.138929	−	0.990302i	$-0.455634\pi$
$191$	3.84008	0.277859	0.138929	+	0.990302i	$0.455634\pi$
$192$	0	0
$193$	−4.02105	−0.289442	−0.144721	−	0.989473i	$-0.546228\pi$
$193$	−4.02105	−0.289442	−0.144721	+	0.989473i	$0.546228\pi$
$194$	0	0
$195$	−3.28432	−0.235195
$196$	0	0
$197$	15.9717	1.13794	0.568969	−	0.822359i	$-0.307343\pi$
$197$	15.9717	1.13794	0.568969	+	0.822359i	$0.307343\pi$
$198$	0	0
$199$	−26.6098	−1.88632	−0.943159	−	0.332343i	$-0.892161\pi$
$199$	−26.6098	−1.88632	−0.943159	+	0.332343i	$0.892161\pi$
$200$	0	0
$201$	−1.13163	−0.0798192
$202$	0	0
$203$	−6.35375	−0.445946
$204$	0	0
$205$	−27.3949	−1.91334
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	24.0129	1.65311	0.826557	−	0.562853i	$-0.190296\pi$
$211$	24.0129	1.65311	0.826557	+	0.562853i	$0.190296\pi$
$212$	0	0
$213$	−8.41595	−0.576651
$214$	0	0
$215$	−1.60415	−0.109402
$216$	0	0
$217$	−9.28432	−0.630260
$218$	0	0
$219$	−5.90952	−0.399328
$220$	0	0
$221$	2.33365	0.156978
$222$	0	0
$223$	−5.17373	−0.346459	−0.173229	−	0.984882i	$-0.555420\pi$
$223$	−5.17373	−0.346459	−0.173229	+	0.984882i	$0.555420\pi$
$224$	0	0
$225$	2.22212	0.148141
$226$	0	0
$227$	26.9507	1.78878	0.894389	−	0.447290i	$-0.147611\pi$
$227$	26.9507	1.78878	0.894389	+	0.447290i	$0.147611\pi$
$228$	0	0
$229$	7.86018	0.519415	0.259708	−	0.965687i	$-0.416374\pi$
$229$	7.86018	0.519415	0.259708	+	0.965687i	$0.416374\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	16.7697	1.09862	0.549310	−	0.835619i	$-0.314891\pi$
$233$	16.7697	1.09862	0.549310	+	0.835619i	$0.314891\pi$
$234$	0	0
$235$	2.93057	0.191169
$236$	0	0
$237$	−2.53472	−0.164648
$238$	0	0
$239$	−26.1445	−1.69115	−0.845573	−	0.533859i	$-0.820741\pi$
$239$	−26.1445	−1.69115	−0.845573	+	0.533859i	$0.820741\pi$
$240$	0	0
$241$	−13.3054	−0.857074	−0.428537	−	0.903524i	$-0.640971\pi$
$241$	−13.3054	−0.857074	−0.428537	+	0.903524i	$0.640971\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.68740	0.171692
$246$	0	0
$247$	−3.66635	−0.233284
$248$	0	0
$249$	10.3537	0.656142
$250$	0	0
$251$	6.30537	0.397991	0.198996	−	0.980000i	$-0.436232\pi$
$251$	6.30537	0.397991	0.198996	+	0.980000i	$0.436232\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	0	0
$255$	−5.13163	−0.321355
$256$	0	0
$257$	7.84008	0.489051	0.244525	−	0.969643i	$-0.421368\pi$
$257$	7.84008	0.489051	0.244525	+	0.969643i	$0.421368\pi$
$258$	0	0
$259$	−11.7285	−0.728777
$260$	0	0
$261$	6.35375	0.393287
$262$	0	0
$263$	−16.1938	−0.998554	−0.499277	−	0.866442i	$-0.666401\pi$
$263$	−16.1938	−0.998554	−0.499277	+	0.866442i	$0.666401\pi$
$264$	0	0
$265$	0.596916	0.0366682
$266$	0	0
$267$	−0.152683	−0.00934403
$268$	0	0
$269$	7.79075	0.475010	0.237505	−	0.971386i	$-0.423670\pi$
$269$	7.79075	0.475010	0.237505	+	0.971386i	$0.423670\pi$
$270$	0	0
$271$	−27.5897	−1.67595	−0.837977	−	0.545706i	$-0.816262\pi$
$271$	−27.5897	−1.67595	−0.837977	+	0.545706i	$0.816262\pi$
$272$	0	0
$273$	1.22212	0.0739658
$274$	0	0
$275$	2.22212	0.133999
$276$	0	0
$277$	−9.06220	−0.544495	−0.272247	−	0.962227i	$-0.587767\pi$
$277$	−9.06220	−0.544495	−0.272247	+	0.962227i	$0.587767\pi$
$278$	0	0
$279$	9.28432	0.555837
$280$	0	0
$281$	0.806169	0.0480920	0.0240460	−	0.999711i	$-0.492345\pi$
$281$	0.806169	0.0480920	0.0240460	+	0.999711i	$0.492345\pi$
$282$	0	0
$283$	8.06220	0.479248	0.239624	−	0.970866i	$-0.422976\pi$
$283$	8.06220	0.479248	0.239624	+	0.970866i	$0.422976\pi$
$284$	0	0
$285$	8.06220	0.477563
$286$	0	0
$287$	10.1938	0.601723
$288$	0	0
$289$	−13.3537	−0.785515
$290$	0	0
$291$	−0.0904840	−0.00530427
$292$	0	0
$293$	−10.7496	−0.627998	−0.313999	−	0.949423i	$-0.601669\pi$
$293$	−10.7496	−0.627998	−0.313999	+	0.949423i	$0.601669\pi$
$294$	0	0
$295$	40.2350	2.34257
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−1.22212	−0.0706768
$300$	0	0
$301$	0.596916	0.0344056
$302$	0	0
$303$	−2.66635	−0.153178
$304$	0	0
$305$	−25.5476	−1.46285
$306$	0	0
$307$	9.42318	0.537809	0.268905	−	0.963167i	$-0.413338\pi$
$307$	9.42318	0.537809	0.268905	+	0.963167i	$0.413338\pi$
$308$	0	0
$309$	5.35375	0.304564
$310$	0	0
$311$	16.7568	0.950193	0.475096	−	0.879934i	$-0.342413\pi$
$311$	16.7568	0.950193	0.475096	+	0.879934i	$0.342413\pi$
$312$	0	0
$313$	−33.5604	−1.89695	−0.948474	−	0.316854i	$-0.897373\pi$
$313$	−33.5604	−1.89695	−0.948474	+	0.316854i	$0.897373\pi$
$314$	0	0
$315$	−2.68740	−0.151418
$316$	0	0
$317$	−32.4499	−1.82257	−0.911283	−	0.411781i	$-0.864907\pi$
$317$	−32.4499	−1.82257	−0.911283	+	0.411781i	$0.864907\pi$
$318$	0	0
$319$	6.35375	0.355742
$320$	0	0
$321$	3.22212	0.179841
$322$	0	0
$323$	−5.72855	−0.318745
$324$	0	0
$325$	−2.71568	−0.150639
$326$	0	0
$327$	−13.4370	−0.743068
$328$	0	0
$329$	−1.09048	−0.0601203
$330$	0	0
$331$	−22.1454	−1.21722	−0.608612	−	0.793468i	$-0.708273\pi$
$331$	−22.1454	−1.21722	−0.608612	+	0.793468i	$0.708273\pi$
$332$	0	0
$333$	11.7285	0.642720
$334$	0	0
$335$	−3.04115	−0.166156
$336$	0	0
$337$	35.4288	1.92993	0.964965	−	0.262378i	$-0.0845068\pi$
$337$	35.4288	1.92993	0.964965	+	0.262378i	$0.0845068\pi$
$338$	0	0
$339$	11.1316	0.604587
$340$	0	0
$341$	9.28432	0.502774
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.68740	0.144685
$346$	0	0
$347$	7.02105	0.376910	0.188455	−	0.982082i	$-0.439652\pi$
$347$	7.02105	0.376910	0.188455	+	0.982082i	$0.439652\pi$
$348$	0	0
$349$	−2.29974	−0.123102	−0.0615511	−	0.998104i	$-0.519605\pi$
$349$	−2.29974	−0.123102	−0.0615511	+	0.998104i	$0.519605\pi$
$350$	0	0
$351$	−1.22212	−0.0652317
$352$	0	0
$353$	−4.79798	−0.255371	−0.127685	−	0.991815i	$-0.540755\pi$
$353$	−4.79798	−0.255371	−0.127685	+	0.991815i	$0.540755\pi$
$354$	0	0
$355$	−22.6170	−1.20039
$356$	0	0
$357$	1.90952	0.101062
$358$	0	0
$359$	−26.9928	−1.42462	−0.712312	−	0.701863i	$-0.752352\pi$
$359$	−26.9928	−1.42462	−0.712312	+	0.701863i	$0.752352\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	−10.0000	−0.524864
$364$	0	0
$365$	−15.8812	−0.831262
$366$	0	0
$367$	−8.88123	−0.463596	−0.231798	−	0.972764i	$-0.574461\pi$
$367$	−8.88123	−0.463596	−0.231798	+	0.972764i	$0.574461\pi$
$368$	0	0
$369$	−10.1938	−0.530670
$370$	0	0
$371$	−0.222116	−0.0115317
$372$	0	0
$373$	3.72855	0.193057	0.0965284	−	0.995330i	$-0.469226\pi$
$373$	3.72855	0.193057	0.0965284	+	0.995330i	$0.469226\pi$
$374$	0	0
$375$	−7.46528	−0.385506
$376$	0	0
$377$	−7.76502	−0.399919
$378$	0	0
$379$	−9.45710	−0.485778	−0.242889	−	0.970054i	$-0.578095\pi$
$379$	−9.45710	−0.485778	−0.242889	+	0.970054i	$0.578095\pi$
$380$	0	0
$381$	−1.06220	−0.0544181
$382$	0	0
$383$	19.0612	0.973984	0.486992	−	0.873406i	$-0.338094\pi$
$383$	19.0612	0.973984	0.486992	+	0.873406i	$0.338094\pi$
$384$	0	0
$385$	−2.68740	−0.136963
$386$	0	0
$387$	−0.596916	−0.0303429
$388$	0	0
$389$	21.2277	1.07629	0.538145	−	0.842852i	$-0.319125\pi$
$389$	21.2277	1.07629	0.538145	+	0.842852i	$0.319125\pi$
$390$	0	0
$391$	−1.90952	−0.0965684
$392$	0	0
$393$	0.555767	0.0280347
$394$	0	0
$395$	−6.81180	−0.342739
$396$	0	0
$397$	−24.2633	−1.21774	−0.608869	−	0.793271i	$-0.708377\pi$
$397$	−24.2633	−1.21774	−0.608869	+	0.793271i	$0.708377\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	−12.2359	−0.611033	−0.305517	−	0.952187i	$-0.598829\pi$
$401$	−12.2359	−0.611033	−0.305517	+	0.952187i	$0.598829\pi$
$402$	0	0
$403$	−11.3465	−0.565210
$404$	0	0
$405$	2.68740	0.133538
$406$	0	0
$407$	11.7285	0.581363
$408$	0	0
$409$	−22.7157	−1.12322	−0.561609	−	0.827403i	$-0.689818\pi$
$409$	−22.7157	−1.12322	−0.561609	+	0.827403i	$0.689818\pi$
$410$	0	0
$411$	−9.56863	−0.471986
$412$	0	0
$413$	−14.9717	−0.736710
$414$	0	0
$415$	27.8247	1.36586
$416$	0	0
$417$	−20.7697	−1.01710
$418$	0	0
$419$	3.43700	0.167908	0.0839542	−	0.996470i	$-0.473245\pi$
$419$	3.43700	0.167908	0.0839542	+	0.996470i	$0.473245\pi$
$420$	0	0
$421$	−9.18002	−0.447407	−0.223703	−	0.974657i	$-0.571815\pi$
$421$	−9.18002	−0.447407	−0.223703	+	0.974657i	$0.571815\pi$
$422$	0	0
$423$	1.09048	0.0530211
$424$	0	0
$425$	−4.24317	−0.205824
$426$	0	0
$427$	9.50643	0.460048
$428$	0	0
$429$	−1.22212	−0.0590043
$430$	0	0
$431$	12.8812	0.620467	0.310234	−	0.950660i	$-0.399593\pi$
$431$	12.8812	0.620467	0.310234	+	0.950660i	$0.399593\pi$
$432$	0	0
$433$	−38.0047	−1.82639	−0.913194	−	0.407525i	$-0.866392\pi$
$433$	−38.0047	−1.82639	−0.913194	+	0.407525i	$0.866392\pi$
$434$	0	0
$435$	17.0751	0.818687
$436$	0	0
$437$	3.00000	0.143509
$438$	0	0
$439$	−26.3958	−1.25981	−0.629903	−	0.776674i	$-0.716905\pi$
$439$	−26.3958	−1.25981	−0.629903	+	0.776674i	$0.716905\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	11.2761	0.535745	0.267872	−	0.963454i	$-0.413679\pi$
$443$	11.2761	0.535745	0.267872	+	0.963454i	$0.413679\pi$
$444$	0	0
$445$	−0.410319	−0.0194510
$446$	0	0
$447$	−18.5476	−0.877270
$448$	0	0
$449$	3.79893	0.179283	0.0896414	−	0.995974i	$-0.471428\pi$
$449$	3.79893	0.179283	0.0896414	+	0.995974i	$0.471428\pi$
$450$	0	0
$451$	−10.1938	−0.480009
$452$	0	0
$453$	8.72855	0.410103
$454$	0	0
$455$	3.28432	0.153971
$456$	0	0
$457$	−5.27050	−0.246544	−0.123272	−	0.992373i	$-0.539339\pi$
$457$	−5.27050	−0.246544	−0.123272	+	0.992373i	$0.539339\pi$
$458$	0	0
$459$	−1.90952	−0.0891286
$460$	0	0
$461$	−20.9928	−0.977731	−0.488865	−	0.872359i	$-0.662589\pi$
$461$	−20.9928	−0.977731	−0.488865	+	0.872359i	$0.662589\pi$
$462$	0	0
$463$	21.9306	1.01920	0.509600	−	0.860411i	$-0.329793\pi$
$463$	21.9306	1.01920	0.509600	+	0.860411i	$0.329793\pi$
$464$	0	0
$465$	24.9507	1.15706
$466$	0	0
$467$	−25.1033	−1.16164	−0.580822	−	0.814030i	$-0.697269\pi$
$467$	−25.1033	−1.16164	−0.580822	+	0.814030i	$0.697269\pi$
$468$	0	0
$469$	1.13163	0.0522539
$470$	0	0
$471$	1.84008	0.0847865
$472$	0	0
$473$	−0.596916	−0.0274462
$474$	0	0
$475$	6.66635	0.305873
$476$	0	0
$477$	0.222116	0.0101700
$478$	0	0
$479$	17.0211	0.777712	0.388856	−	0.921299i	$-0.372870\pi$
$479$	17.0211	0.777712	0.388856	+	0.921299i	$0.372870\pi$
$480$	0	0
$481$	−14.3337	−0.653558
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−0.243167	−0.0110416
$486$	0	0
$487$	22.2149	1.00665	0.503326	−	0.864096i	$-0.332109\pi$
$487$	22.2149	1.00665	0.503326	+	0.864096i	$0.332109\pi$
$488$	0	0
$489$	3.41595	0.154474
$490$	0	0
$491$	2.47252	0.111583	0.0557916	−	0.998442i	$-0.482232\pi$
$491$	2.47252	0.111583	0.0557916	+	0.998442i	$0.482232\pi$
$492$	0	0
$493$	−12.1326	−0.546424
$494$	0	0
$495$	2.68740	0.120790
$496$	0	0
$497$	8.41595	0.377507
$498$	0	0
$499$	−10.6874	−0.478434	−0.239217	−	0.970966i	$-0.576891\pi$
$499$	−10.6874	−0.478434	−0.239217	+	0.970966i	$0.576891\pi$
$500$	0	0
$501$	−10.3748	−0.463512
$502$	0	0
$503$	−36.8941	−1.64503	−0.822513	−	0.568746i	$-0.807429\pi$
$503$	−36.8941	−1.64503	−0.822513	+	0.568746i	$0.807429\pi$
$504$	0	0
$505$	−7.16555	−0.318863
$506$	0	0
$507$	−11.5064	−0.511019
$508$	0	0
$509$	−10.9095	−0.483556	−0.241778	−	0.970332i	$-0.577731\pi$
$509$	−10.9095	−0.483556	−0.241778	+	0.970332i	$0.577731\pi$
$510$	0	0
$511$	5.90952	0.261422
$512$	0	0
$513$	3.00000	0.132453
$514$	0	0
$515$	14.3877	0.633996
$516$	0	0
$517$	1.09048	0.0479594
$518$	0	0
$519$	14.6591	0.643464
$520$	0	0
$521$	−24.0257	−1.05259	−0.526293	−	0.850303i	$-0.676419\pi$
$521$	−24.0257	−1.05259	−0.526293	+	0.850303i	$0.676419\pi$
$522$	0	0
$523$	−8.64625	−0.378074	−0.189037	−	0.981970i	$-0.560537\pi$
$523$	−8.64625	−0.378074	−0.189037	+	0.981970i	$0.560537\pi$
$524$	0	0
$525$	−2.22212	−0.0969811
$526$	0	0
$527$	−17.7285	−0.772268
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	14.9717	0.649717
$532$	0	0
$533$	12.4580	0.539618
$534$	0	0
$535$	8.65911	0.374366
$536$	0	0
$537$	10.3255	0.445577
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−4.77065	−0.205106	−0.102553	−	0.994728i	$-0.532701\pi$
$541$	−4.77065	−0.205106	−0.102553	+	0.994728i	$0.532701\pi$
$542$	0	0
$543$	6.65911	0.285770
$544$	0	0
$545$	−36.1106	−1.54681
$546$	0	0
$547$	−9.43700	−0.403497	−0.201748	−	0.979437i	$-0.564662\pi$
$547$	−9.43700	−0.403497	−0.201748	+	0.979437i	$0.564662\pi$
$548$	0	0
$549$	−9.50643	−0.405725
$550$	0	0
$551$	19.0612	0.812036
$552$	0	0
$553$	2.53472	0.107787
$554$	0	0
$555$	31.5193	1.33792
$556$	0	0
$557$	22.8319	0.967418	0.483709	−	0.875229i	$-0.339289\pi$
$557$	22.8319	0.967418	0.483709	+	0.875229i	$0.339289\pi$
$558$	0	0
$559$	0.729500	0.0308546
$560$	0	0
$561$	−1.90952	−0.0806198
$562$	0	0
$563$	2.33365	0.0983517	0.0491758	−	0.998790i	$-0.484341\pi$
$563$	2.33365	0.0983517	0.0491758	+	0.998790i	$0.484341\pi$
$564$	0	0
$565$	29.9151	1.25854
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−12.1033	−0.507399	−0.253699	−	0.967283i	$-0.581647\pi$
$569$	−12.1033	−0.507399	−0.253699	+	0.967283i	$0.581647\pi$
$570$	0	0
$571$	35.4653	1.48418	0.742088	−	0.670303i	$-0.233836\pi$
$571$	35.4653	1.48418	0.742088	+	0.670303i	$0.233836\pi$
$572$	0	0
$573$	3.84008	0.160422
$574$	0	0
$575$	2.22212	0.0926687
$576$	0	0
$577$	−27.4992	−1.14481	−0.572403	−	0.819972i	$-0.693989\pi$
$577$	−27.4992	−1.14481	−0.572403	+	0.819972i	$0.693989\pi$
$578$	0	0
$579$	−4.02105	−0.167109
$580$	0	0
$581$	−10.3537	−0.429546
$582$	0	0
$583$	0.222116	0.00919912
$584$	0	0
$585$	−3.28432	−0.135790
$586$	0	0
$587$	21.8684	0.902604	0.451302	−	0.892371i	$-0.350960\pi$
$587$	21.8684	0.902604	0.451302	+	0.892371i	$0.350960\pi$
$588$	0	0
$589$	27.8529	1.14766
$590$	0	0
$591$	15.9717	0.656989
$592$	0	0
$593$	−12.4442	−0.511023	−0.255512	−	0.966806i	$-0.582244\pi$
$593$	−12.4442	−0.511023	−0.255512	+	0.966806i	$0.582244\pi$
$594$	0	0
$595$	5.13163	0.210376
$596$	0	0
$597$	−26.6098	−1.08907
$598$	0	0
$599$	−3.75055	−0.153243	−0.0766217	−	0.997060i	$-0.524413\pi$
$599$	−3.75055	−0.153243	−0.0766217	+	0.997060i	$0.524413\pi$
$600$	0	0
$601$	28.7131	1.17123	0.585616	−	0.810588i	$-0.300853\pi$
$601$	28.7131	1.17123	0.585616	+	0.810588i	$0.300853\pi$
$602$	0	0
$603$	−1.13163	−0.0460836
$604$	0	0
$605$	−26.8740	−1.09258
$606$	0	0
$607$	−33.6098	−1.36418	−0.682089	−	0.731269i	$-0.738929\pi$
$607$	−33.6098	−1.36418	−0.682089	+	0.731269i	$0.738929\pi$
$608$	0	0
$609$	−6.35375	−0.257467
$610$	0	0
$611$	−1.33270	−0.0539152
$612$	0	0
$613$	−10.1728	−0.410875	−0.205437	−	0.978670i	$-0.565862\pi$
$613$	−10.1728	−0.410875	−0.205437	+	0.978670i	$0.565862\pi$
$614$	0	0
$615$	−27.3949	−1.10467
$616$	0	0
$617$	31.7907	1.27985	0.639924	−	0.768439i	$-0.278966\pi$
$617$	31.7907	1.27985	0.639924	+	0.768439i	$0.278966\pi$
$618$	0	0
$619$	−5.13982	−0.206587	−0.103293	−	0.994651i	$-0.532938\pi$
$619$	−5.13982	−0.206587	−0.103293	+	0.994651i	$0.532938\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	0.152683	0.00611710
$624$	0	0
$625$	−31.1728	−1.24691
$626$	0	0
$627$	3.00000	0.119808
$628$	0	0
$629$	−22.3958	−0.892981
$630$	0	0
$631$	39.7285	1.58157	0.790784	−	0.612095i	$-0.209673\pi$
$631$	39.7285	1.58157	0.790784	+	0.612095i	$0.209673\pi$
$632$	0	0
$633$	24.0129	0.954426
$634$	0	0
$635$	−2.85455	−0.113279
$636$	0	0
$637$	−1.22212	−0.0484220
$638$	0	0
$639$	−8.41595	−0.332930
$640$	0	0
$641$	39.1866	1.54778	0.773889	−	0.633322i	$-0.218309\pi$
$641$	39.1866	1.54778	0.773889	+	0.633322i	$0.218309\pi$
$642$	0	0
$643$	32.2981	1.27371	0.636857	−	0.770982i	$-0.280234\pi$
$643$	32.2981	1.27371	0.636857	+	0.770982i	$0.280234\pi$
$644$	0	0
$645$	−1.60415	−0.0631634
$646$	0	0
$647$	−2.20107	−0.0865328	−0.0432664	−	0.999064i	$-0.513776\pi$
$647$	−2.20107	−0.0865328	−0.0432664	+	0.999064i	$0.513776\pi$
$648$	0	0
$649$	14.9717	0.587691
$650$	0	0
$651$	−9.28432	−0.363881
$652$	0	0
$653$	−9.97990	−0.390544	−0.195272	−	0.980749i	$-0.562559\pi$
$653$	−9.97990	−0.390544	−0.195272	+	0.980749i	$0.562559\pi$
$654$	0	0
$655$	1.49357	0.0583586
$656$	0	0
$657$	−5.90952	−0.230552
$658$	0	0
$659$	16.5347	0.644101	0.322051	−	0.946722i	$-0.395628\pi$
$659$	16.5347	0.644101	0.322051	+	0.946722i	$0.395628\pi$
$660$	0	0
$661$	−14.4169	−0.560752	−0.280376	−	0.959890i	$-0.590459\pi$
$661$	−14.4169	−0.560752	−0.280376	+	0.959890i	$0.590459\pi$
$662$	0	0
$663$	2.33365	0.0906315
$664$	0	0
$665$	−8.06220	−0.312639
$666$	0	0
$667$	6.35375	0.246018
$668$	0	0
$669$	−5.17373	−0.200028
$670$	0	0
$671$	−9.50643	−0.366992
$672$	0	0
$673$	47.9142	1.84696	0.923478	−	0.383651i	$-0.125333\pi$
$673$	47.9142	1.84696	0.923478	+	0.383651i	$0.125333\pi$
$674$	0	0
$675$	2.22212	0.0855293
$676$	0	0
$677$	−36.2350	−1.39262	−0.696312	−	0.717740i	$-0.745177\pi$
$677$	−36.2350	−1.39262	−0.696312	+	0.717740i	$0.745177\pi$
$678$	0	0
$679$	0.0904840	0.00347246
$680$	0	0
$681$	26.9507	1.03275
$682$	0	0
$683$	−37.3666	−1.42979	−0.714897	−	0.699230i	$-0.753526\pi$
$683$	−37.3666	−1.42979	−0.714897	+	0.699230i	$0.753526\pi$
$684$	0	0
$685$	−25.7147	−0.982509
$686$	0	0
$687$	7.86018	0.299885
$688$	0	0
$689$	−0.271452	−0.0103415
$690$	0	0
$691$	−17.6663	−0.672060	−0.336030	−	0.941851i	$-0.609084\pi$
$691$	−17.6663	−0.672060	−0.336030	+	0.941851i	$0.609084\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−55.8165	−2.11724
$696$	0	0
$697$	19.4653	0.737300
$698$	0	0
$699$	16.7697	0.634288
$700$	0	0
$701$	39.6226	1.49653	0.748263	−	0.663402i	$-0.230888\pi$
$701$	39.6226	1.49653	0.748263	+	0.663402i	$0.230888\pi$
$702$	0	0
$703$	35.1856	1.32705
$704$	0	0
$705$	2.93057	0.110372
$706$	0	0
$707$	2.66635	0.100278
$708$	0	0
$709$	14.8181	0.556505	0.278252	−	0.960508i	$-0.410245\pi$
$709$	14.8181	0.556505	0.278252	+	0.960508i	$0.410245\pi$
$710$	0	0
$711$	−2.53472	−0.0950593
$712$	0	0
$713$	9.28432	0.347700
$714$	0	0
$715$	−3.28432	−0.122826
$716$	0	0
$717$	−26.1445	−0.976384
$718$	0	0
$719$	−34.2149	−1.27600	−0.638000	−	0.770036i	$-0.720238\pi$
$719$	−34.2149	−1.27600	−0.638000	+	0.770036i	$0.720238\pi$
$720$	0	0
$721$	−5.35375	−0.199384
$722$	0	0
$723$	−13.3054	−0.494832
$724$	0	0
$725$	14.1188	0.524358
$726$	0	0
$727$	33.5265	1.24343	0.621715	−	0.783243i	$-0.286436\pi$
$727$	33.5265	1.24343	0.621715	+	0.783243i	$0.286436\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.13982	0.0421578
$732$	0	0
$733$	−11.1938	−0.413454	−0.206727	−	0.978399i	$-0.566281\pi$
$733$	−11.1938	−0.413454	−0.206727	+	0.978399i	$0.566281\pi$
$734$	0	0
$735$	2.68740	0.0991262
$736$	0	0
$737$	−1.13163	−0.0416842
$738$	0	0
$739$	25.4508	0.936223	0.468112	−	0.883669i	$-0.344935\pi$
$739$	25.4508	0.936223	0.468112	+	0.883669i	$0.344935\pi$
$740$	0	0
$741$	−3.66635	−0.134687
$742$	0	0
$743$	15.4982	0.568575	0.284288	−	0.958739i	$-0.408243\pi$
$743$	15.4982	0.568575	0.284288	+	0.958739i	$0.408243\pi$
$744$	0	0
$745$	−49.8448	−1.82617
$746$	0	0
$747$	10.3537	0.378824
$748$	0	0
$749$	−3.22212	−0.117734
$750$	0	0
$751$	24.4241	0.891249	0.445625	−	0.895220i	$-0.352982\pi$
$751$	24.4241	0.891249	0.445625	+	0.895220i	$0.352982\pi$
$752$	0	0
$753$	6.30537	0.229780
$754$	0	0
$755$	23.4571	0.853691
$756$	0	0
$757$	−44.9079	−1.63221	−0.816103	−	0.577907i	$-0.803870\pi$
$757$	−44.9079	−1.63221	−0.816103	+	0.577907i	$0.803870\pi$
$758$	0	0
$759$	1.00000	0.0362977
$760$	0	0
$761$	36.9918	1.34095	0.670476	−	0.741931i	$-0.266090\pi$
$761$	36.9918	1.34095	0.670476	+	0.741931i	$0.266090\pi$
$762$	0	0
$763$	13.4370	0.486452
$764$	0	0
$765$	−5.13163	−0.185535
$766$	0	0
$767$	−18.2972	−0.660673
$768$	0	0
$769$	−2.66730	−0.0961854	−0.0480927	−	0.998843i	$-0.515314\pi$
$769$	−2.66730	−0.0961854	−0.0480927	+	0.998843i	$0.515314\pi$
$770$	0	0
$771$	7.84008	0.282354
$772$	0	0
$773$	−11.9918	−0.431316	−0.215658	−	0.976469i	$-0.569190\pi$
$773$	−11.9918	−0.431316	−0.215658	+	0.976469i	$0.569190\pi$
$774$	0	0
$775$	20.6308	0.741081
$776$	0	0
$777$	−11.7285	−0.420759
$778$	0	0
$779$	−30.5815	−1.09570
$780$	0	0
$781$	−8.41595	−0.301146
$782$	0	0
$783$	6.35375	0.227064
$784$	0	0
$785$	4.94504	0.176496
$786$	0	0
$787$	29.5969	1.05502	0.527508	−	0.849550i	$-0.323126\pi$
$787$	29.5969	1.05502	0.527508	+	0.849550i	$0.323126\pi$
$788$	0	0
$789$	−16.1938	−0.576516
$790$	0	0
$791$	−11.1316	−0.395795
$792$	0	0
$793$	11.6180	0.412566
$794$	0	0
$795$	0.596916	0.0211704
$796$	0	0
$797$	26.6509	0.944024	0.472012	−	0.881592i	$-0.343528\pi$
$797$	26.6509	0.944024	0.472012	+	0.881592i	$0.343528\pi$
$798$	0	0
$799$	−2.08230	−0.0736664
$800$	0	0
$801$	−0.152683	−0.00539478
$802$	0	0
$803$	−5.90952	−0.208542
$804$	0	0
$805$	−2.68740	−0.0947184
$806$	0	0
$807$	7.79075	0.274247
$808$	0	0
$809$	−18.9023	−0.664569	−0.332284	−	0.943179i	$-0.607819\pi$
$809$	−18.9023	−0.664569	−0.332284	+	0.943179i	$0.607819\pi$
$810$	0	0
$811$	−33.2277	−1.16678	−0.583392	−	0.812191i	$-0.698275\pi$
$811$	−33.2277	−1.16678	−0.583392	+	0.812191i	$0.698275\pi$
$812$	0	0
$813$	−27.5897	−0.967612
$814$	0	0
$815$	9.18002	0.321562
$816$	0	0
$817$	−1.79075	−0.0626503
$818$	0	0
$819$	1.22212	0.0427042
$820$	0	0
$821$	19.0550	0.665023	0.332511	−	0.943099i	$-0.392104\pi$
$821$	19.0550	0.665023	0.332511	+	0.943099i	$0.392104\pi$
$822$	0	0
$823$	−27.2350	−0.949352	−0.474676	−	0.880161i	$-0.657435\pi$
$823$	−27.2350	−0.949352	−0.474676	+	0.880161i	$0.657435\pi$
$824$	0	0
$825$	2.22212	0.0773642
$826$	0	0
$827$	−34.8730	−1.21265	−0.606327	−	0.795215i	$-0.707358\pi$
$827$	−34.8730	−1.21265	−0.606327	+	0.795215i	$0.707358\pi$
$828$	0	0
$829$	−50.4216	−1.75121	−0.875607	−	0.483024i	$-0.839538\pi$
$829$	−50.4216	−1.75121	−0.875607	+	0.483024i	$0.839538\pi$
$830$	0	0
$831$	−9.06220	−0.314364
$832$	0	0
$833$	−1.90952	−0.0661608
$834$	0	0
$835$	−27.8812	−0.964870
$836$	0	0
$837$	9.28432	0.320913
$838$	0	0
$839$	−15.3886	−0.531274	−0.265637	−	0.964073i	$-0.585582\pi$
$839$	−15.3886	−0.531274	−0.265637	+	0.964073i	$0.585582\pi$
$840$	0	0
$841$	11.3701	0.392073
$842$	0	0
$843$	0.806169	0.0277659
$844$	0	0
$845$	−30.9224	−1.06376
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	8.06220	0.276694
$850$	0	0
$851$	11.7285	0.402049
$852$	0	0
$853$	−26.5121	−0.907756	−0.453878	−	0.891064i	$-0.649960\pi$
$853$	−26.5121	−0.907756	−0.453878	+	0.891064i	$0.649960\pi$
$854$	0	0
$855$	8.06220	0.275721
$856$	0	0
$857$	51.2972	1.75228	0.876139	−	0.482058i	$-0.160111\pi$
$857$	51.2972	1.75228	0.876139	+	0.482058i	$0.160111\pi$
$858$	0	0
$859$	−13.4314	−0.458272	−0.229136	−	0.973394i	$-0.573590\pi$
$859$	−13.4314	−0.458272	−0.229136	+	0.973394i	$0.573590\pi$
$860$	0	0
$861$	10.1938	0.347405
$862$	0	0
$863$	35.1938	1.19801	0.599006	−	0.800745i	$-0.295563\pi$
$863$	35.1938	1.19801	0.599006	+	0.800745i	$0.295563\pi$
$864$	0	0
$865$	39.3949	1.33947
$866$	0	0
$867$	−13.3537	−0.453517
$868$	0	0
$869$	−2.53472	−0.0859844
$870$	0	0
$871$	1.38299	0.0468607
$872$	0	0
$873$	−0.0904840	−0.00306242
$874$	0	0
$875$	7.46528	0.252373
$876$	0	0
$877$	−46.4910	−1.56989	−0.784945	−	0.619566i	$-0.787309\pi$
$877$	−46.4910	−1.56989	−0.784945	+	0.619566i	$0.787309\pi$
$878$	0	0
$879$	−10.7496	−0.362575
$880$	0	0
$881$	33.5815	1.13139	0.565695	−	0.824615i	$-0.308608\pi$
$881$	33.5815	1.13139	0.565695	+	0.824615i	$0.308608\pi$
$882$	0	0
$883$	44.5805	1.50025	0.750127	−	0.661293i	$-0.229992\pi$
$883$	44.5805	1.50025	0.750127	+	0.661293i	$0.229992\pi$
$884$	0	0
$885$	40.2350	1.35248
$886$	0	0
$887$	−46.3738	−1.55708	−0.778541	−	0.627594i	$-0.784040\pi$
$887$	−46.3738	−1.55708	−0.778541	+	0.627594i	$0.784040\pi$
$888$	0	0
$889$	1.06220	0.0356250
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	3.27145	0.109475
$894$	0	0
$895$	27.7486	0.927535
$896$	0	0
$897$	−1.22212	−0.0408053
$898$	0	0
$899$	58.9902	1.96743
$900$	0	0
$901$	−0.424135	−0.0141300
$902$	0	0
$903$	0.596916	0.0198641
$904$	0	0
$905$	17.8957	0.594873
$906$	0	0
$907$	52.9507	1.75820	0.879099	−	0.476639i	$-0.158145\pi$
$907$	52.9507	1.75820	0.879099	+	0.476639i	$0.158145\pi$
$908$	0	0
$909$	−2.66635	−0.0884372
$910$	0	0
$911$	−12.3054	−0.407695	−0.203847	−	0.979003i	$-0.565345\pi$
$911$	−12.3054	−0.407695	−0.203847	+	0.979003i	$0.565345\pi$
$912$	0	0
$913$	10.3537	0.342659
$914$	0	0
$915$	−25.5476	−0.844577
$916$	0	0
$917$	−0.555767	−0.0183530
$918$	0	0
$919$	21.2277	0.700239	0.350119	−	0.936705i	$-0.386141\pi$
$919$	21.2277	0.700239	0.350119	+	0.936705i	$0.386141\pi$
$920$	0	0
$921$	9.42318	0.310504
$922$	0	0
$923$	10.2853	0.338544
$924$	0	0
$925$	26.0622	0.856920
$926$	0	0
$927$	5.35375	0.175840
$928$	0	0
$929$	−2.82627	−0.0927268	−0.0463634	−	0.998925i	$-0.514763\pi$
$929$	−2.82627	−0.0927268	−0.0463634	+	0.998925i	$0.514763\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	0	0
$933$	16.7568	0.548594
$934$	0	0
$935$	−5.13163	−0.167822
$936$	0	0
$937$	4.77693	0.156056	0.0780278	−	0.996951i	$-0.475138\pi$
$937$	4.77693	0.156056	0.0780278	+	0.996951i	$0.475138\pi$
$938$	0	0
$939$	−33.5604	−1.09520
$940$	0	0
$941$	42.0257	1.37000	0.685000	−	0.728543i	$-0.259802\pi$
$941$	42.0257	1.37000	0.685000	+	0.728543i	$0.259802\pi$
$942$	0	0
$943$	−10.1938	−0.331957
$944$	0	0
$945$	−2.68740	−0.0874211
$946$	0	0
$947$	56.5944	1.83907	0.919535	−	0.393009i	$-0.128566\pi$
$947$	56.5944	1.83907	0.919535	+	0.393009i	$0.128566\pi$
$948$	0	0
$949$	7.22212	0.234440
$950$	0	0
$951$	−32.4499	−1.05226
$952$	0	0
$953$	−21.7276	−0.703826	−0.351913	−	0.936033i	$-0.614469\pi$
$953$	−21.7276	−0.703826	−0.351913	+	0.936033i	$0.614469\pi$
$954$	0	0
$955$	10.3198	0.333942
$956$	0	0
$957$	6.35375	0.205388
$958$	0	0
$959$	9.56863	0.308987
$960$	0	0
$961$	55.1985	1.78060
$962$	0	0
$963$	3.22212	0.103831
$964$	0	0
$965$	−10.8062	−0.347863
$966$	0	0
$967$	34.5686	1.11165	0.555826	−	0.831299i	$-0.312402\pi$
$967$	34.5686	1.11165	0.555826	+	0.831299i	$0.312402\pi$
$968$	0	0
$969$	−5.72855	−0.184027
$970$	0	0
$971$	−5.12535	−0.164480	−0.0822402	−	0.996613i	$-0.526207\pi$
$971$	−5.12535	−0.164480	−0.0822402	+	0.996613i	$0.526207\pi$
$972$	0	0
$973$	20.7697	0.665846
$974$	0	0
$975$	−2.71568	−0.0869715
$976$	0	0
$977$	9.11877	0.291735	0.145868	−	0.989304i	$-0.453403\pi$
$977$	9.11877	0.291735	0.145868	+	0.989304i	$0.453403\pi$
$978$	0	0
$979$	−0.152683	−0.00487976
$980$	0	0
$981$	−13.4370	−0.429010
$982$	0	0
$983$	−24.3795	−0.777584	−0.388792	−	0.921325i	$-0.627108\pi$
$983$	−24.3795	−0.777584	−0.388792	+	0.921325i	$0.627108\pi$
$984$	0	0
$985$	42.9224	1.36762
$986$	0	0
$987$	−1.09048	−0.0347105
$988$	0	0
$989$	−0.596916	−0.0189808
$990$	0	0
$991$	10.3547	0.328928	0.164464	−	0.986383i	$-0.447411\pi$
$991$	10.3547	0.328928	0.164464	+	0.986383i	$0.447411\pi$
$992$	0	0
$993$	−22.1454	−0.702765
$994$	0	0
$995$	−71.5111	−2.26705
$996$	0	0
$997$	5.14545	0.162958	0.0814790	−	0.996675i	$-0.474036\pi$
$997$	5.14545	0.162958	0.0814790	+	0.996675i	$0.474036\pi$
$998$	0	0
$999$	11.7285	0.371075

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.a.j.1.3	✓	3
3.2	odd	2		5796.2.a.o.1.1		3
4.3	odd	2		7728.2.a.br.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.j.1.3	✓	3	1.1	even	1	trivial
5796.2.a.o.1.1		3	3.2	odd	2
7728.2.a.br.1.3		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1932.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.68740 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.68740 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.22212 q^{13} +2.68740 q^{15} -1.90952 q^{17} +3.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +2.22212 q^{25} +1.00000 q^{27} +6.35375 q^{29} +9.28432 q^{31} +1.00000 q^{33} -2.68740 q^{35} +11.7285 q^{37} -1.22212 q^{39} -10.1938 q^{41} -0.596916 q^{43} +2.68740 q^{45} +1.09048 q^{47} +1.00000 q^{49} -1.90952 q^{51} +0.222116 q^{53} +2.68740 q^{55} +3.00000 q^{57} +14.9717 q^{59} -9.50643 q^{61} -1.00000 q^{63} -3.28432 q^{65} -1.13163 q^{67} +1.00000 q^{69} -8.41595 q^{71} -5.90952 q^{73} +2.22212 q^{75} -1.00000 q^{77} -2.53472 q^{79} +1.00000 q^{81} +10.3537 q^{83} -5.13163 q^{85} +6.35375 q^{87} -0.152683 q^{89} +1.22212 q^{91} +9.28432 q^{93} +8.06220 q^{95} -0.0904840 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - q^{13} + q^{15} + 4 q^{17} + 9 q^{19} - 3 q^{21} + 3 q^{23} + 4 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 3 q^{33} - q^{35} + 6 q^{37} - q^{39} + 3 q^{41} + 15 q^{43} + q^{45} + 13 q^{47} + 3 q^{49} + 4 q^{51} - 2 q^{53} + q^{55} + 9 q^{57} + 14 q^{59} - 2 q^{61} - 3 q^{63} + 14 q^{65} + 9 q^{67} + 3 q^{69} + 11 q^{71} - 8 q^{73} + 4 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 16 q^{83} - 3 q^{85} + 4 q^{87} + 11 q^{89} + q^{91} + 4 q^{93} + 3 q^{95} - 10 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - q^13 + q^15 + 4 * q^17 + 9 * q^19 - 3 * q^21 + 3 * q^23 + 4 * q^25 + 3 * q^27 + 4 * q^29 + 4 * q^31 + 3 * q^33 - q^35 + 6 * q^37 - q^39 + 3 * q^41 + 15 * q^43 + q^45 + 13 * q^47 + 3 * q^49 + 4 * q^51 - 2 * q^53 + q^55 + 9 * q^57 + 14 * q^59 - 2 * q^61 - 3 * q^63 + 14 * q^65 + 9 * q^67 + 3 * q^69 + 11 * q^71 - 8 * q^73 + 4 * q^75 - 3 * q^77 - 12 * q^79 + 3 * q^81 + 16 * q^83 - 3 * q^85 + 4 * q^87 + 11 * q^89 + q^91 + 4 * q^93 + 3 * q^95 - 10 * q^97 + 3 * q^99

Introduction

L-functions

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